A novel adaptive multi-scale wavelet Galerkin method for solving fuzzy hybrid differential equations

Area of Science:

• Numerical analysis
• Applied mathematics
• Computational science

Background:

• Fuzzy hybrid differential equations (FHDEs) model complex systems with uncertainty and mixed behaviors.
• Traditional numerical methods struggle with FHDEs due to nonlinearity, discontinuities, and fuzzy parameters.
• Accurate and efficient solutions for FHDEs are crucial in control engineering, biology, and economics.

Purpose of the Study:

• Introduce a novel numerical scheme, the Adaptive Multiwavelet Galerkin (AMWG) method, for solving FHDEs.
• Address the limitations of existing methods in handling uncertainty and sharp transitions in FHDEs.
• Demonstrate the AMWG method's accuracy, efficiency, and scalability for complex dynamical systems.

Main Methods:

• Combines wavelet-based multi-resolution analysis with the Galerkin projection technique.
• Employs local error estimates for adaptive refinement of the solution domain.
• Applies selective refinement: fine for steep gradients/discontinuities, coarse elsewhere.

Main Results:

• The AMWG method significantly reduces computational cost without sacrificing accuracy.
• Achieves higher accuracy, lower memory requirements, and faster computation compared to traditional methods.
• Effectively handles fuzzy uncertainty and sharp transitions in benchmark FHDEs.

Conclusions:

• The AMWG method is a powerful, flexible, and scalable numerical tool for FHDEs.
• Offers superior performance for complex dynamical systems with nonlinearity and discrete switching.
• Has significant potential for large-scale scientific and engineering applications.